The Pro League outfit sacked previous incumbent Emilio Ferrera after losing the opening game of the season 3-1 to Mechelen.
Genk have drawn their three matches since then and have now moved to appoint a man who counts two Scottish Premier League titles and an English League Cup crown among his honours as a coach.
It marks the end of an 18-month absence from the game for the 55-year-old, who left Nottingham Forest by mutual consent in February 2013 after just seven games at the helm.
McLeish, who has signed a two-year deal, will seek to restore the three-times Belgian champions to their former glory, with the club's last title having come as recently as 2011.
The Scot enjoyed a hugely successful playing career in his homeland, collecting three top-flight titles and a European Cup Winners Cup medal with Aberdeen.